For monads $S$ and $T$ on a fixed Abelian category $C$, a morphism of monads $\sigma: S\rightarrow T$ induces a functor between Eilenberg-Moore categories $\sigma^*:C^T\rightarrow C^S$. This functor sends a $T$-algebra $(A,\rho:TA\rightarrow A)$ to an $S$-algebra $(A,\rho\circ \sigma_A:SA\rightarrow A)$. This functor commutes with forgetful functors $U^T:C^T\rightarrow C$ and $U^S:C^S\rightarrow C$.

Are there any known results that would also provide a left adjoint $\sigma_!$ to $\sigma^*$? In addition, I would like this left adjoint to commute with the free algebra functors for $T$ and $S$, i.e. $F^T=\sigma_!\circ F^S$.

Does it even make sense to expect such a functor to exist?

The reason I'm looking for a left adjoint is that I want to work with a category of Abelian categories whose morphisms are pairs of adjoint functors. Monads $S$ and $T$ are induced by an adjoint pair, there is a canonical $free \dashv forgetful$ adjunction for EM categories, so I would like to have an adjoint for $\sigma^*$ too.